Second Fundamental Theorem of Calculus
- Author:
- Dr. Jack L. Jackson II
Accumulation Function
In this activity we start with a function f(x) which is graphed in the left window. Enter the desired formula in the input box for f(x). For now set a = 0, x = 0, and C = 0 via their sliders or input boxes. Now slide the slider for x slowly to the right.
The shaded area is accumulating as we increase the value for x. Green areas accumulate positively and red areas accumulate negatively. The green area minus the red area is the value of the integral of f(x) over
the interval [a, x]. This is demonstrated by the size of the vertical line segment in the window on the right. Note that this defines a new accumulation function A. Note that this function goes through the
origin.
Check the checkbox for A(x) in the right window. You will now see the entire accumulation function. On this blue accumulation function which is graphed in the right window, y-values are equal to the green area -
red area from the window on the left.
If we adjust the value of a, then the accumulation will now go through (a, 0) instead of the origin.
If we adjust the value of C, then the entire accumulation function will be shifted vertically by C units.
Note that this gives us a way to take any function we can integrate and use it to define a new related accumulation function. Actually, there are a whole family of these accumulation functions for different choices of a and C.
Second Fundamental Theorem of Calculus
(Fundamental Theorem of Calculus Part 2)
Click on the A'(x) checkbox in the right window. This will graph the derivative of the accumulation function in red in the right window.
How does A'(x) compare to the original f(x)?
They are the same! This illustrates the
Second Fundamental Theorem of Calculus
For any function f which is continuous on the interval containing a, x, and all values between them:
This tells us that each of these accumulation functions are antiderivatives of the original function f.
First integrating and then differentiating returns you back to the original function. In this sense, integration and differentiation are inverse processes.